Feedback linearization-based tracking control of a tilt-rotor with cat-trot gait plan

With the introduction of the laterally bounded forces, the tilt-rotor gains more flexibility in the controller design. Typical feedback linearization methods utilize all the inputs in controlling this vehicle; the magnitudes as well as the directions of the thrusts are maneuvered simultaneously based on a unified control rule. Although several promising results indicate that these controllers may track the desired complicated trajectories, the tilting angles are required to change relatively fast or in large scale during the flight, which turns to be a challenge in application. The recent gait plan for a tilt-rotor may solve this problem; the tilting angles are fixed or vary in a predetermined pattern without being maneuvered by the control algorithm. Carefully avoiding the singular decoupling matrix, several attitudes can be tracked without changing the tilting angles frequently. While the position was not directly regulated in that research, which left the position-tracking still an open question. In this research, we elucidate the coupling relationship between the position and the attitude. Based on this, we design the position-tracking controller, adopting feedback linearization. A cat-trot gait is further designed for a tilt-rotor to track the reference; three types of references are designed for our tracking experiments: set point, uniform rectilinear motion, and uniform circular motion. The significant improvement with less steady state error is witnessed after equipping with our modified attitude–position decoupler. It is also found that the frequency of the cat-trot gait highly influenced the steady state error.

Cat-Inspired Gaits for a Tilt-Rotor—From Symmetrical to Asymmetrical

Among the tilt-rotors (quadrotors) developed in recent decades, Ryll’s model with eight inputs (four magnitudes of thrusts and four tilting angles) attracted great attention. Typical feedback linearization maneuvers all of the eight inputs with a united control rule to stabilize this tilt-rotor. Instead of assigning the tilting angles by the control rule, the recent research predetermines the tilting angles and leaves the magnitudes of thrusts with the only control signals. These tilting angles are designed to mimic the cat-trot gait while avoiding the singular decoupling matrix in feedback linearization. To complete the discussions of the cat-gait inspired tilt-rotor gaits, this research addresses the analyses on the rest of the common cat gaits, walk, run, transverse gallop, and rotary gallop. It is found that the singular decoupling matrix exists in walk gait, transverse gallop gait, and rotary gallop gait; the decoupling matrix can hardly be guaranteed to be invertible analytically. Further modifications (scaling) are conducted to these three gaits to accommodate the application of feedback linearization; the acceptable attitudes, leading to invertible decoupling matrix, for each scaled gait are evaluated in the roll-pitch diagram. The modified gaits with different periods are then applied to the tilt-rotor in tracking experiments, in which the references are uniform rectilinear motion and uniform circular motion with or without the equipment of the modified attitude-position decoupler. All the experiments are simulated in Simulink, MATLAB. The result shows that these gaits, after modifications, are feasible in tracking references, especially for the cases equipped with the modified attitude-position decoupler.

Four-Dimensional Gait Surfaces for a Tilt-Rotor—Two Color Map Theorem

This article presents the four-dimensional surfaces that guide the gait plan for a tilt-rotor. The previous gaits analyzed in the tilt-rotor research are inspired by animals; no theoretical base backs the robustness of these gaits. This research deduces the gaits by diminishing the adverse effect of the attitude of the tilt-rotor for the first time. Four-dimensional gait surfaces are subsequently found on which the gaits are expected to be robust to the attitude. These surfaces provide the region where the gait is suggested to be planned. However, a discontinuous region may hinder the gait plan process while utilizing the proposed gait surfaces. The ‘Two Color Map Theorem’ is then established to guarantee the continuity of each gait designed. The robustness of the typical gaits on the gait surface, obeying the Two Color Map Theorem, is demonstrated by comparing the singular curves in attitude with the gaits not on the gait surface. The result shows that the gaits on the gait surface receive wider regions of the acceptable attitudes.

Singular Zone in Quadrotor Yaw–Position Feedback Linearization

Feedback linearization-based controllers are widely exploited in stabilizing a tilt rotor (eight or twelve inputs); each degree of freedom (six degrees of freedom in total) is manipulated individually to track the desired trajectory, since no singular decoupling matrix is introduced while applying this method. The conventional quadrotor (four inputs), on the other hand, is an under-actuated MIMO system that can directly track four independent degrees of freedom at most. Common selections of these outputs can be yaw–position and attitude–altitude. It is reported that no singularity is found in the decoupling matrix while applying feedback linearization in the yaw–position-tracking problem. However, in this research, we argue the existence of the ignored singular zone within the range of interest, which can cause the failure in the controller design. This paper visualizes this noninvertible area and details the process of deduction for the first time. An attempt (switch controller) to avert the singular problem is later discussed with the verification by simulation in Simulink and MATLAB. All the results are sketched in the roll–pitch diagram.